Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX-XXX
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Thermodynamic Consistency Test for Experimental Hydrogen Solubility Data in Alkenes Mohammad Jamali,† Amir Abbas Izadpanah,*,† and Masoud Mofarahi†,‡ †
Department of Chemical Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, Iran ‡ Department of Chemical and Biomolecular Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, Republic of Korea ABSTRACT: In this communication, a thermodynamic consistency test is performed on the isothermal solubility of hydrogen in alkenes and vapor−liquid equilibrium data for systems containing hydrogen and alkenes. The improved Peng−Robinson equation of state was used for this purpose. The fugacity coefficient of each component and compressibility factor of liquid and vapor phases were calculated by this equation of state. Thermodynamic consistency tests have been performed by rewriting the Gibbs−Duhem equation in terms fugacity coefficient and using the area test method. The results showed that most of the equilibrium data of the liquid phase were consistent, but for the vapor phase less than one-half of the data was consistent.
1. INTRODUCTION Hydrogen is widely used in many industries for a variety of applications. Hydrogen solubility in various hydrocarbon solutions is one of the factors needed to design and optimize the operation of processes such as hydrogenation, hydrocracking, and hydrotreatment. Hydrogen is used extensively in refineries and in petrochemical complexes; therefore, the solubility of hydrogen in the hydrocarbons used in these industries is an important property. Alkenes are among the hydrocarbons that exist in the olefin units of the petrochemical industry. In the olefin unit of the petrochemicals complex, ethylene, propylene, and 1-butene, etc. are produced. Except for ethylene, other alkenes can be produced as some unwanted and side products. In petrochemical plants, some of the alkenes are heavier than ethylene that hydrogenated and converted into alkane and are returned to the cracking furnace. Therefore, hydrogen solubility in olefins can be of great importance, applicable to the petrochemical industry and should be investigated. Equilibrium data for hydrogen solubility in some of the alkenes have been presented, but so far, the thermodynamic consistency of these data has not been reported. Because these data are used for practical and theoretical purposes, their thermodynamic consistency is an important indicator of the reliability and accuracy of such data. Sagara et al. empirically reported the solubility and vapor− liquid phase equilibrium of hydrogen in ethylene.1 In another work, they also reported the equilibrium data on hydrogen solubility in propylene.2 Vasilva et al. studied and reported the solubility of hydrogen in 1-butene.3 Sokolov and Polyakov have investigated the solubility of hydrogen in 1-hexene and 1heptene.4 Hydrogen and 1-octene systems were also studied by Sokulov and Poliakov as well as by Xie et al.4,5 © XXXX American Chemical Society
The thermodynamic consistency test of experimental data is based on the Gibbs−Dohem equation.6−10 This equation expresses the relationship between the activity or fugacity coefficient of a species in a solution and in one phase. Bertucco et al.11 rewrote the Gibbs−Duhem equation at constant temperature in terms of fugacity coefficient, and presented a method for the thermodynamic consistency test for vapor− liquid equilibrium data at high pressures. Valderama et al. also used this method to study the thermodynamic consistency of high-pressure gas−liquid equilibrium data, equilibrium data of solid solubility in a gas phase at high-pressure, the ternary system of solid solubility in the gas phase at high pressure, vapor−liquid equilibrium of mixtures containing ionic liquids as well as high-pressure vapor−liquid equilibrium data, for both vapor and liquid phases.12−16 Eslamimanesh et al. also used this method to study the thermodynamic consistency of experimental data for water content of methane,17 the amount of sulfur with hydrogen sulfide,18 the solubility of carbon dioxide and methane in water inside and outside the hydrate formation region,19 and the phase behavior of supercritical carbon dioxide and ionic liquids.20 Mohammadi et al. used the same method to study the thermodynamic consistency of experimental data on the solubility of wax in the gas phase21 and the glycol loss in a gaseous system.22 Recently, Kondori et al. have developed this method to study the thermodynamic consistency of experReceived: August 12, 2017 Accepted: October 23, 2017
A
DOI: 10.1021/acs.jced.7b00729 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
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Table 1. Critical Properties and Acentric Factors hydrogen
ethylene
propylene
1-butene
1-heptene
1-hexene
1-octene
33.19 13.13 −0.216 9
282.30 50.40 0.087 9
365.60 46.65 0.140 9
420.00 40.43 0.191 9
537.29 28.30 0.331 9
504.00 31.40 0.280 9
567.00 26.80 0.393 6
Tc (K) Pc (bar) ω ref
imental data for water content of methane at constant pressure.23 In this work, the thermodynamic consistency test for the solubility of hydrogen in alkenes and the vapor−liquid equilibrium of these systems are investigated by using the improved Peng−Robinson (PR) equation of state24 and the thermodynamic consistency test method at constant temperature.
Aϕ = 1
A ϕ2 = =
∑ xidM̅ i = 0
(2)
For a binary system this equation can be written as (3)
or ⎛ 1 ⎞ ⎛ 1 ⎞ ⎟ d ln(ϕ ̂ ) ⎟ dP = ⎜ ⎜ 1 ⎝ Z − 1⎠ ⎝ x1P ⎠ +
⎛ 1 ⎞ (1 − x1) ⎟ d ln(ϕ2̂ ) ⎝ Z − 1 ⎠ x1
⎜
(4)
Equation 5 can be written in the integral form as ⎛
⎞
⎛
⎞
∫ ⎜⎝ x1P ⎟⎠ dP = ∫ ⎜⎜ (Z −11)ϕ̂ ⎟⎟ d(ϕ1̂ ) ⎝
1
+
∫
1⎠
⎛ ⎞ 1 − x1 ⎜⎜ ⎟⎟ d(ϕ ̂ ) 2 ⎝ x1(Z − 1)(ϕ2̂ ) ⎠
(5)
In eq 5, the left-hand side is shown by Ap ⎞
⎛
1
1 ⎞ ⎟ dP ⎜ ⎝ x1P ⎠
P(2) ⎛
∫ ⎜⎝ x1P ⎟⎠ dP = A p = ∫P(1)
(6)
and the right-hand side is shown by Aϕ
Aϕ = A ϕ1 + A ϕ2
⎞ 1 − x1 ⎜⎜ ⎟⎟ d(ϕ ̂ ) 2 ⎝ x1(Z − 1)(ϕ2̂ ) ⎠
(9)
|Aϕ − AP| Ap
(10)
3. RESULTS AND DISCUSSIONS Owing to the success of the PR equation with kij obtained by the GCM method25 to model various systems, including hydrogen and alkenes systems,26 we initially intended to use this method to predict the hydrogen solubility in the alkenes. But according to the results obtained from the GCM method for calculating the bubble point pressure, which in many of these systems the percentage error in calculating bubble point pressure was more than 10%, it was decided to use the PR equation with van der Waals one-fluid mixing rule and fitted kij. The results showed that by fitting the solubility data and obtaining kij in this way, this method can well model these systems. Therefore, this equation of state (PR with fitted kjj) was used to calculate the hydrogen solubility in the alkenes and then do the thermodynamic consistency test of equilibrium data. In this work, the thermodynamic consistency test was done for experimental data in both the liquid and vapor phases. First, modeling of the vapor−liquid phase equilibrium and obtaining the hydrogen solubility in alkenes using the improved PR equation of state was done, andthen the kij between hydrogen and alkenes was evaluated. Then thermodynamic consistency test was done on the equilibrium data, and for this purpose the compressibility factor of the vapor and liquid phases and fugacity coefficient of components in the vapor and liquid phases were calculated using this equation of state. Table 1 presents the critical properties of the compounds used. Binary interaction parameters between hydrogen and alkenes, kij, have been calculated by fitting the equation of state on the experimental data of hydrogen solubility in alkenes. The binary interaction coefficients between hydrogen and the alkenes have been evaluated by minimizing the absolute average error between the calculated bubble point pressure and the experimental data.
nc
⎛ Z − 1⎞ ⎜ ⎟ dP − x d ln(ϕ ̂ ) − x d ln(ϕ ̂ ) = 0 1 2 1 2 ⎝ P ⎠
ϕ2̂ (2) ⎛
∫ϕ̂ (1)
The values of the fugacity coefficient and the compressibility factor are calculated using the improved PR equation of state.24
⎛G R ⎞ ∑ xid⎜ i̅ ⎟ ⎝ RT ⎠ i=1
i=1
∫
ΔA% = 100
nc
∑ xi d ln(ϕi)̂ = 0
1
⎞ 1 ⎜⎜ ⎟⎟ d(ϕ ̂ ) 1 ⎝ (Z − 1)ϕ1̂ ⎠
⎛ ⎞ 1 − x1 ⎜⎜ ⎟⎟ d(ϕ ̂ ) 2 ⎝ x1(Z − 1)(ϕ2̂ ) ⎠
(1)
If M is equal to G /(RT), the Gibbs−Duhem equation at constant temperature is written as
⎛ VR ⎞ =⎜ ⎟ dP − ⎝ RT ⎠
1⎠
ϕ1̂ (2) ⎛
If the percent area error between experimental (Ap) and calculated area (Aϕ) calculated by eq 10 is less than 20%, the experimental data will be considered thermodynamically consistent.12
R
⎛ ⎡ GR ⎤ ⎞ ⎜ ∂⎣⎢ RT ⎦⎥ ⎟ ⎜ ⎟ dP − ⎜ ∂P ⎟ ⎝ ⎠T , x̲
⎝
2
nc
i=1
⎞
(8)
2. EQUATIONS The Gibbs−Duhem equation that expresses the relation between partial molar properties of species in one phase is given as ⎛ ∂M ⎞ ⎛ ∂M ⎞ ⎜ ⎟ ⎟ dT + ⎜ dP − ⎝ ∂T ⎠ P , x̲ ⎝ ∂P ⎠T , x̲
⎛
∫ ⎜⎜ (Z −11)ϕ̂ ⎟⎟ d(ϕ1̂ ) = ∫ϕ̂ (1)
(7) B
DOI: 10.1021/acs.jced.7b00729 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
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Table 2. Results of the Thermodynamic Consistency Test of Data Related to the Hydrogen−Alkene System in the Liquid Phase T (K)
a
kij
Ap
Aϕ1
Aϕ2
123.15 148.15 173.15 198.15 223.15
0.1010 0.1265 0.1690 0.1898 0.2315
182.62 128.18 103.62 85.720 87.693
1. 6327 1.6215 1.6104 1.6106 1.4144
173.20 198.20 223.20 248.20
0.2200 0.3200 0.3649 0.3220
197.21 155.08 112.57 92.971
1.6807 1.6476 1.6146 1.6149
253.20 314.69 356.78
0.0055 0.2341 0.3877
25.452 23.391 18.492
122.60 3.5902 2.0254
333.20 393.20 403.20 453.20
0.2003 0.3557 0.3552 0.5084
31.926 6.0512 28.184 4.5535
4.6827 2.6277 937.24 2.4484
333.20 403.20
0.3444 0.3907
42.220 28.959
8.9872 4.9104
313.15 328.20 333.15 393.20 463.20
0.1471 0.3095 0.2227 0.3505 0.4827
145.42 47.738 165.21 31.749 23.915
2.6948 17.580 2.6979 7.4469 4.9323
ΔA%
Aϕ
H2 + Ethylene 158.31 109. 60 86.007 69.203 64.725 H2 + Propylene 161.68 131.56 96.807 77.415 H2 + 1-Butene 65.673 16.825 13.450 H2 + 1-Hexene 18.046 3.5557 1921.3 2.4124 H2 + 1-Heptene 25.619 15.467 H2 + 1-Octene 108.80 23.931 105.75 18.105 12.127
ΔP%
resulta
ref
159. 95 111. 22 87. 618 70.814 66.140
12.42 13.23 15.44 17.39 24.58
7.86 5.34 5.22 4.97 1.09
TC TC TC TC TI
1 1 1 1 1
163.36 133.21 98.422 79.030
17.16 14.10 12.57 15.00
4.71 5.12 5.08 4.17
TC TC TC TC
2 2 2 2
188.27 20.416 15.504
639.73 12.72 16.16
1.62 2.41 1.44
TI TC TC
3 3 3
28.18 2.18 10042 6.75
3.92 1.82 3.81 4.31
TI TC TI TC
4 4 4 4
34.607 20.377
18.03 29.63
6.34 2.69
TC TI
4 4
111.50 41.511 108.45 25.552 17.060
23.33 13.04 34.36 19.52 28.67
1.38 8.67 1.41 3.57 4.68
TI TC TI TC TI
5 4 5 4 4
22.729 6.1834 2858.5 4.8508
TC = thermodynamically consistent; TI = thermodynamically inconsistent.
Table 3. Results of the Thermodynamic Consistency Test of Data Related to the Hydrogen−Alkene System in the Vapor Phase T (K)
a
kij
Ap
Aϕ1
123.15 148.15 173.15 198.15 223.15
0.1010 0.1265 0.1690 0.1898 0.2315
1.4628 1.4820 1.5419 1.7608 2.4799
1.3361 0.2766 2.2469 0.5160 0.4777
173.15 198.20 223.20 248.20
0.2200 0.3200 0.3649 0.3220
1.4583 1.4660 1.4983 1.6084
2.1100 1.7820 2.8524 4.5138
393.20 453.20
0.3557 0.5084
0.9730 1.0565
0.8303 20.624
313.15 333.15
0.1471 0.2227
2.2353 2.2811
2.3936 2.4792
Aϕ2 H2 + Ethylene 0.1460 1.0427 3.6359 2.2426 1.8907 H2 + Propylene 0 0.1954 0.9753 5.0655 H2 + 1-Hexene 0.1814 168.52 H2 + 1-Octene 0.1404 0.2630
Aϕ
ΔA%
1.4821 1.3194 5.8829 2.7586 2.3684
resulta
ref
1.32 10.97 281.53 56.67 4.50
TC TC TI TI TC
1 1 1 1 1
2.1100 1.9765 3.8278 9.5803
44.69 34.83 155.48 495.63
TI TI TI TI
2 2 2 2
1.0117 189.14
3.97 17803
TC TI
4 4
2.5340 2.6421
13.36 15.83
TC TC
5 5
TC = thermodynamically consistent; TI = thermodynamically inconsistent.
ΔP % =
N |P expi − Pcal i| ⎛ 100 ⎞ ⎜ ⎟∑ ⎝ N ⎠ Pexp i=1
i
the parameters of thermodynamic models. Among them, the simulated annealing27 method has been used to minimize the objective function in this work. The algorithm used in the simulated annealing to find the minimum of a function is a stochastic method. This method usually converges to a global minimum. If the obtained kij does not satisfy the average absolute error less than 10% for
(11)
where N is the number of experimental points, Pcal is the calculated bubble point pressure by the improved PR equation of state, and Pexp is experimental pressure. Eslamimanesh et al.18 have listed some of the optimization methods used to obtain C
DOI: 10.1021/acs.jced.7b00729 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
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Table 4. Sample of the Calculated Values of Various Quantities in Liquid and Vapor Phases for the Hydrogen (1)/Ethylene (2) System at 123.15 K x1
y1
P (bar)
Z
ϕ̂ 1
ϕ̂ 2
1/(Px1)
1/[(Z − 1)ϕ̂ 1]
0.9960 0.9970 0.9980 0.9980
20.27 40.53 60.80 81.06 20.27 40.53 60.80 81.06
0.0814 0.1623 0.2428 0.3230 0.9865 0.9812 0.9833 0.9909
201.88 104.91 72.816 57.096 0.9859 0.9856 0.9690 0.9657
0.0012 0.0007 0.0005 0.0004 0.7180 0.5318 0.4073 0.3204
9.2031 2.6473 1.2952 0.8450 0.0495 0.0247 0.0165 0.0124
−0.0054 −0.0114 −0.0181 −0.0259 −74.965 −54.444 −61.720 −113.17
0.0053 0.0093 0.0127 0.0146
Figure 1. Behavior of functions (a) 1/Px1 vs P, (b) 1/[(Z − 1)ϕ̂ 1] vs ϕ1, and (c)
1 − x1 vs x1(Z − 1)ϕ2̂
1 − x1 x1(Z − 1)ϕ2̂
−167149 −192257 −213881 −255614 −0.4134 −0.3006 −0.2943 −0.6836
AP
Aϕ1
Aϕ2
120.99 39.947 21.685
0.8131 0.4737 0.3459
100.63 36.552 21.127
0.7528 0.4178 0.2923
0.6613 0.3845 0.2903
0.0665 0.0370 0.0425
ϕ2 in the liquid phase for the system hydrogen (1) and
ethylene (2) at the temperature 123.15 K.
modeling bubble point pressure or solubility, these parameters cannot be used for the thermodynamic consistency test. The results show that this equation of state can correlate experimental data with an average absolute error of less than 9% at th ebubble point pressure. Therefore, this model can be used to test the thermodynamic consistency of these data. The kijvalues obtained at each temperature and the average absolute error in calculating the bubble point pressure are given in Table 2. In addition, the results for calculating areas AP and Aϕ for the liquid phase are given in this table. Integration has been done to obtain areas using the trapezoidal rule. According to Table 2, of the 23 equilibrium data reported for the liquid phase, 8, about 34% of the data, do not have thermodynamic consistency. Hydrogen and propylene data are thermodynamically consistent in all temperatures, but equilibrium data for the hydrogen and 1-octene system are not thermodynamically consistent at three temperatures. Table 3 also shows the thermodynamic consistency test for experimental data of hydrogen and some of the alkenes systems in the vapor phase. The results show that more than half the experimental equilibrium data of hydrogen and alkenes in the vapor phase are not thermodynamically consistent and experimental data for a system such as hydrogen and propylene are not thermodynamically consistent for all studied temperatures. According to Tables 2 and 3 for the hydrogen and 1-hexene systems, there is a significant deviation in ΔA% for the liquid phase at 403.2 and for the gas phase at 453.2 K. According to the calculations carried out, in one of the equilibrium points (for liquid phase at 294.2 bar and hydrogen mole fraction equal
to 0.303 and for gas phase at pressure of 100 bar and hydrogen mole fraction equal to 0.8), the Z value was very close to 1 (Z for the liquid phase was 0.999825 and for the gas phase was 1 and 1 − x1 0.999801). Therefore, the values of the (Z − 1)ϕ1
x1(Z − 1)ϕ2
were increased unusually and the amount of the calculated Aϕ area was large. This causes a large deviation in ΔA% for this system at these temperatures. If this point (where the Z value for this point is very close to 1) is eliminated from the calculation, then the obtained value for the ΔA% will be acceptable (for the liquid phase ΔA% will be equal to 31.05 and for the gas phase equal to 44.45). Valderrama and Alvarez12 and Valderrama and Faundez13 reported a high error for the ΔA% at some equilibrium data points. Despite the fact that the ΔP% and Δy% for these points were in the usual range and below 10%, a large error was seen in the ΔA%, and that reason was not mentioned and discussed. There are two integrals in the Aϕ equation which contain in their denominator (Z − 1). If conditions are such that the calculated compressibility factor of the mixture is close to 1, then the values of these integrals will increase abnormally and there will be a lot of difference between ΔA% values. A sample calculation and its results for AP and Aϕ at both liquid and vapor phases, for hydrogen and the ethylene system at 123.15 K are presented in Table 4. For example, the behavior of functions (a) 1/Px1 vs P, (b) 1/ [(Z − 1)ϕ̂ 1] vs ϕ1, and (c) 1 − x1 ̂ vs ϕ2 in the liquid phase x1(Z − 1)ϕ2
for the system hydrogen (1) and ethylene (2) at the temperature 123.15 K is also plotted in Figure 1. D
DOI: 10.1021/acs.jced.7b00729 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
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4. CONCLUSION The improved PR equation of state was used to describe the vapor and liquid phases of systems including alkenes and hydrogen. The results showed that this model could well correlate the solubility of hydrogen in alkenes. Using this equation of state, the thermodynamic consistency test was performed on the equilibrium data of the liquid and vapor phases of these systems. The results showed that about 65% of the liquid phase data was thermodynamically consistent with acceptable accuracy, but more than 50% of the vapor phase data was not thermodynamically consistent. The results showed that equilibrium data measurements for these systems in the gas phase should be done more carefully so that these data can be used to adjust the parameters of thermodynamic models and other applications.
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REFERENCES
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Tel.: +98 77 33222600. Fax: +98 77 3344 1495. ORCID
Amir Abbas Izadpanah: 0000-0003-2051-5119 Masoud Mofarahi: 0000-0001-8583-7923 Notes
The authors declare no competing financial interest.
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Article
LIST OF SYMBOLS
Symbols
kij = binary interaction parameter P = pressure Pc = critical pressure R = ideal gas constant T = temperature Tc = critical temperature x1 = experimental solubility of hydrogen in the alkenes y1 = experimental mole fraction of hydrogen in vapor phase Z = compressibility factor G̅ Ri = residual Gibbs free energy VR = residual volume ln = natural logarithm N = number of data points Abbreviations
EoS = equation of state PR = Peng−Robinson TC = thermodynamic consistent TI = thermodynamic inconsistent A = area AP = experimental area Aϕ = calculation area Δ% = percent deviation ΔA% = area deviation ΔP% = pressure deviation Greek letters
ω = acentric factor φ = fugacity coefficient Super/subscripts
cal = calculated exp = experimental i, j = components i and j E
DOI: 10.1021/acs.jced.7b00729 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
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(23) Kondori, J.; Javanmardi, J.; Eslamimanesh, A.; Mohammadi, A. H. Thermodynamic consistency test for isobaric experimental data of water content of methane. Fluid Phase Equilib. 2013, 347, 54−61. (24) Robinson, D. B., Peng, D. Y. The Characterization of the Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs; GPA Research Report/PR:28; Gas Processors Association, 1978. (25) Jaubert, J.-N.; Mutelet, F. VLE predictions with the Peng− Robinson equation of state and temperature dependent kij calculated through a group contribution method. Fluid Phase Equilib. 2004, 224, 285−304. (26) Qian, J.-W.; Jaubert, J.-N.; Privat, R. Prediction of the phase behavior of alkene-containing binary systems with the PPR78 model. Fluid Phase Equilib. 2013, 354, 212−235. (27) Kirkpatrick, S.; Gelatt, C. D.; Vecchi, M. P. Optimization by simulated annealing. Science 1983, 220, 671−680.
F
DOI: 10.1021/acs.jced.7b00729 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
